Explain xkcd: It's 'cause you're dumb.
Cueball explains to Megan that the gravity on Titan is far lighter than the gravity on Earth, allowing him to fly given the appropriate technology. Cueball has already put together the apparatus necessary to test this: by weighing himself down sufficiently, he can recreate the effects of Titan's gravity. He uses this to fly around, expressing his delight at being able to do so, only Megan points out a fatal flaw: his wings use hot glue, which can easily be melted by a large heat source, and Black Hat is waiting nearby with a large heat source. Naturally, Cueball falls.
Titan is the largest moon of Saturn. Icarus was a character in Greek mythology who is most famous for the fatal end of his flight, when the wax holding the wings together melted and he fell to his death in what became known as the Icarian Sea. In the original, this was because Icarus ignored instructions not to fly too close to the Sun (often taken as tragic examples of hubris); here Black Hat is bringing an artificial "sun" (the heat lamp) to "Icarus" to recreate the tragedy.
- Cueball: Titan's gravity is 14% of Earth's, and its atmosphere 50% denser.
- Cueball: So if you can generate 9% of your body weight in lift, you can fly on Titan.
- Cueball: With wings, a stage harness, a cable, and 91% of my weight in bricks, I want to test this.
- [There is a heap of materials on the ground. Cueball is holding a stage harness.]
- [Large diagram of a bridge. A rope leads through pulleys tied to the bridge. One end goes to Cueball, one end to a pile of bricks.]
- [Cueball is standing with wings attached to his arms.]
- [Cueball flaps the wings, and appears to be floating.]
- [Cueball glides.]
- Cueball: It works!
- Megan: Except you have two problems.
- Cueball: What?
- Megan: You used hot glue on your wing joints and you have friends into Greek mythology.
- Cueball: Huh?
- [Black Hat is standing on the bridge, with a large lamp labeled "heat lamp" attached to a battery.]
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- [The wing segments fall off Cueball and he tumbles downward.]
Cueball's physics has a mistake on this one (or at least assumes we've managed to heat the atmosphere of Titan to Earth's temperature). The temperature of Titan is roughly 1/3 the temperature of Earth on an absolute scale. Starting with the Ideal Gas Law, PV = NkT (k is Boltzmann's constant, N is # of molecules, P is pressure, V is volume, T is temperature), its easy to define the density of a gas, ρ as:
ρ = m/V = (m P)/(N k T) = P (m/N) / (k T)
Titan's atmosphere is 98.4% molecular nitrogen (N2) and on Earth only 78.1% molecular nitrogen (by volume), but for simplicity we'll assume 100% for both. The weight of one molecule of Nitrogen is (m/N) ~ 2 × 14 × 1.67x10-27 (kg/molecule) (there are 28 nucleons per molecule with a mass of about 1.67x10^-27 kg.
The pressure on Titan is PTitan=146.7 kPa, and TTitan = 93.7 K, while on Earth PEarth=101.3 kPa and TEarth = 287 K.
Plugging in numbers, we get ρTitan = 5.3 kg/m3 and ρEarth = 1.2 kg/m3 (note the measured surface density of air on Earth is 1.2 kg/m3 at Earth's mean temperature even without the simplifying assumption of 100% N2).
Hence Titan's atmosphere is 4.4 = (5.3/1.2) times denser than Earth's (or 340% denser); not 50% denser as stated in the comic.
You will get the 50% denser if you assume the same planetary temperature on Titan as on Earth. Titan at 287 K would have a density of ρTitan at 287K ~ 1.73 kg/m3 which is about 50% greater than Earth's.
For the second calculation (panel 2), note lift is proportional to the density of air. If your action on Earth creates a lift of L0 and you weigh W0, on Titan you'd have a lift of 4.4 L0 (Cueball calculated 1.5 L0) due to the greater air density. Your weight would only be 0.14 W0, due to Titan's lower surface gravity. If lift balances weight, you would be able to fly on Titan, that is if 4.4 L0 = 0.14 W0. That means to fly on Titan you need a lift on Earth of L0 = 0.03 W0, that is 3% of your Earth weight. Substituting Cueball's Titan density you would get the critical value from the comic: L0 = 0.14 W0/(1.5) = 9% W0.
PS: I largely adapted this my writeup on xkcd forums from 2009 when the comic was made
) 05:44, 8 June 2013 (UTC)